Channeled spectropolarimetry refers to analysis of polarization states of an electromagnetic (EM) signal as a function of respective wavelengths of the EM signal. A conventional channeled spectropolarimeter (CSP) comprises a plurality of crystal retarders, which are typically composed of a uniaxial crystal, such as calcite or quartz (e.g., for the visible near infrared (VISNIR) spectrum). Other exemplary uniaxial crystals include cadmium sulfide and cadmium selenide (e.g., for the thermal infrared spectrum (LWIR)). These uniaxial crystals have two distinct indices of refraction corresponding to two separate crystal axes (e.g., a fast axis and a slow axis) in the uniaxial crystal, wherein the crystal axes are orthogonal to one another.
The crystal retarders are typically designed for a particular application, such as characterization of a particular object, discriminating between objects (e.g., man-made or natural), etc. Thicknesses of crystal retarders in a conventional CSP are selected based upon desired amplitude modulating carrier frequencies respectively induced by the crystal retarders in an EM signal being analyzed. Orientations of the crystal retarders (relative to one another) are selected based upon polarization states that are modulated onto the EM signal passing through the crystal retarder. Thus, a crystal retarder in a CSP amplitude modulates the frequency spectrum of the EM signal passing therethrough with a particular carrier frequency and certain polarization states. The amplitude modulating carrier frequency is based upon a thickness of the crystal retarder, and the orientation of the fast axis and the slow axis in the crystal retarder relative to the transmission axis of the EM signal define the polarization states that are amplitude modulated onto a frequency spectrum of the EM signal. Accordingly, to obtain an accurate measurement for the frequency spectrum of the EM signal that is subject to analysis, crystal retarders in a CSP must have a relatively precise orientation.
A primary limitation of conventional channeled CSP is related to the sensitivity of the crystal retarders to temperature. With more particularity, a conventional CSP includes two uniaxial crystal retarders arranged optically in series with one another and oriented in a particular manner relative to one another. In such a conventional configuration, the crystal retarders act to amplitude modulate the desirably measured frequency spectrum with carrier frequencies that comprise the spectrally-dependent Stokes parameters S1, S2, and S3. Using a particular configuration, intensity measured by the spectrometer of the CSP is as follows:
                              I          ⁡                      (            σ            )                          =                                            1              2                        ⁡                          [                                                                                                                                            S                          0                                                ⁡                                                  (                          σ                          )                                                                    +                                                                                                    S                            1                                                    ⁡                                                      (                            σ                            )                                                                          ⁢                                                  cos                          ⁡                                                      (                                                          ϕ                              2                                                        )                                                                                              +                                                                                                                                                                                                                S                            2                                                    ⁡                                                      (                            σ                            )                                                                          ⁢                                                  sin                          ⁡                                                      (                                                          ϕ                              1                                                        )                                                                          ⁢                                                  sin                          ⁡                                                      (                                                          ϕ                              2                                                        )                                                                                              +                                                                                                    S                            3                                                    ⁡                                                      (                            σ                            )                                                                          ⁢                                                  cos                          ⁡                                                      (                                                          ϕ                              1                                                        )                                                                          ⁢                                                  sin                          ⁡                                                      (                                                          ϕ                              2                                                        )                                                                                                                                                          ]                                .                                    (        1        )            
In Eq. 1, φi(σ)=2πσBli is the phase difference introduced by the ith retarder, B=|n1−n2| is the birefringence of the uniaxial crystal, li is the crystal retarder thickness, and σ=1/λ. To recover the state of polarization (SOP) information based upon the output of the spectrometer, Stokes information must be separated from the phase information associated with the modulating carrier frequencies (φ1 and φ2), which are produced by the crystal retarders of the CSP. One technique used to accomplish this involves acquiring a reference measurement using a known SOP, thereby allowing for the phase terms to be characterized. This approach for obtaining a measurement using a CSP can be referred to as the reference beam calibration technique.
A limitation of conventional CSPs and the reference beam calibration technique relates to the temperature sensitivity of the crystal retarders. With more specificity, when sample data is acquired at a temperature that is different from the temperature when the calibration data was acquired, calibration errors are introduced into the resultant measurement. Temperature change produces a variation in thickness and dispersion of the birefringent retarder elements, resulting in a change in the carrier frequencies in the modulated spectral measurements. The change in phase of the ith retarder for a change of temperature ΔT is given by the following algorithm:
                                          Δϕ            i                    ≈                      2            ⁢            πσ            ⁢                                                  ⁢                          l              i                        ⁢            Δ            ⁢                                                  ⁢                          T              ⁡                              [                                                      B                    ⁡                                          (                      σ                      )                                                        ⁢                                                            γ                      L                                        ⁡                                          (                                                                                                    ∂                                                          n                              1                                                                                                            ∂                            T                                                                          -                                                                              ∂                                                          n                              2                                                                                                            ∂                            T                                                                                              )                                                                      ]                                                    ,                            (        2        )            where γL=(1/li)(∂li/∂T) is the coefficient of linear thermal expansion along the propagation direction of the EM signal.
Eq. (2) implies that when the calibration data is applied after the instrument experiences a change in temperature, the carrier frequency phases are not effectively compensated. This produces calibration errors in the polarization data products. To avoid these calibration errors, calibration data must be taken frequently, or precise thermal stability of the system must be maintained actively. For many CSPs, such as those that are field deployed, these solutions complicate system operation significantly.